König's theorem (set theory)

In set theory, König's theorem states that if the axiom of choice holds, I is a set, ${\displaystyle \kappa _{i}}$ and ${\displaystyle \lambda _{i}}$ are cardinal numbers for every i in I, and ${\displaystyle \kappa _{i}<\lambda _{i}}$ for every i in I, then

${\displaystyle \sum _{i\in I}\kappa _{i}<\prod _{i\in I}\lambda _{i}.}$

The sum here is the cardinality of the disjoint union of the sets m_i, and the product is the cardinality of the Cartesian product. However, without the use of the axiom of choice, the sum and the product cannot be defined as cardinal numbers, and the meaning of the inequality sign would need to be clarified.

König's theorem was introduced by König (1904) in the slightly weaker form that the sum of a strictly increasing sequence of nonzero cardinal numbers is less than their product.